numerical simulations of dense suspensions rheology …

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III International Conference on Particle-based Methods – Fundamentals and ApplicationsPARTICLES 2013

M. Bischoff, E. Onate, D.R.J. Owen, E. Ramm & P. Wriggers (Eds)

NUMERICAL SIMULATIONS OF DENSE SUSPENSIONSRHEOLOGY USING A DEM-FLUID COUPLED MODEL

D.MARZOUGUI1, B.CHAREYRE1 AND J.CHAUCHAT2

1 Grenoble INP, UJF, UMR CNRS 5521, 3SR Lab.BP 53, 38041, Grenoble cedex 9, France

[email protected]

2 Grenoble INP, UJF, UMR CNRS 5521, 3SR Lab.BP 53, 38041, Grenoble cedex 9, France

[email protected]

2 Grenoble INP, UJF, UMR CNRS 5519, LEGIBP 53, 38041, Grenoble cedex 9, France

[email protected]

Key words: DEM, PFV, suspension, lubrication, hydromechanical coupling, viscosity

Abstract. The understanding of dense suspensions rheology is of great practical interestfor both industrial and geophysical applications and has led to a large amount of publica-tions over the past decades. This problem is especially difficult as it is a two-phase mediain which particle-particle interactions as well as fluid-particle interactions are significant.In this contribution, the plane shear flow of a dense fluid-grain mixture is studied usingthe DEM-PFV coupled model. We further improve the original model: including thedeviatoric part of the stress tensor on the basis of the lubrication theory, and extendingthe solver to periodic boundary conditions. Simulations of a granular media saturatedby an incompressible fluid and subjected to a plane shear at imposed vertical stress arepresented. The shear stress is decomposed in different contributions which can be ex-amined separately: contact forces, lubrication forces and drag forces associated to theporomechanical couplings.

1 INTRODUCTION

The rheology of grain-fluid mixtures is subject of practical interest for both industrialand geophysical applications. When the solid fraction of such mixture is high enough,i.e. in dense suspensions, the bulk behavior is affected by intricated phenomena combin-ing the viscosity of the fluid phase as well as the interactions between the solid particlesthrough solid contacts. Moreover, the contact interactions may be modified by the pres-ence of the fluid, as described by lubrication theories. Additionaly, in transient situations,

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Numerical simulation of dense suspension rheology using a DEM-fluid coupled model

125

D.Marzougui, B.Chareyre and J.Chauchat

poromechanical couplings may develop long range interactions by coupling the local rateof volume change to the pore pressure field.

Direct particle-scale modeling of this problem is a promising way to better evaluatethe interactions between phases and to link the microscale properties and phenomena tothe quantities measured for the bulk material, as it needs much less simplifications thanformer analytical developements (such as [6, 11, 1]). This modeling can be based on lubri-cation models [9], or more elaborated methods to reflect the fluid viscosity through pairinteractions between particles [14]. This is advantageous as it does not need to actuallysolve Navier-Stokes (NS) equations in the fluid phase. The price to pay is that long rangeinteractions due to poromechanical couplings are difficult to reflect. An alternative isto really solve NS in the fluid phase using a CFD solver, or to use a lattice-Boltzmanmodel [7]. It is to be noted that direct resolution of NS does not eliminate the need fora proper modeling of the lubrication forces, due to mesh size dependencies [8]. The maindifficulty associated to this approach is the high computational cost, so that followinglarge deformations of thousands of immersed particles in 3D remains a challenging task.

A new method to simulate fluid-particle interactions has been developped recentlyand may be of some help to tackle the computational challenge [3]. In this method, thesolid phase is modelized with the discrete element method (DEM), and the fluid flow issolved using a pore-scale finite volume method (PFV). The key aspects of this DEM-PFV coupling are recalled in the first part of this paper. It was implemented in theopen source code Yade-DEM [12]. Extensions of this method in order to study densesuspensions are being undertaken by the first author. Namely, the original model lacks acoupling term to link the fluid forces to the deviatoric strain, as explained in section 2.2.We also generalized the boundary conditions in order to allow very large deformations ofthe suspension in simple shear. Typical results of the preliminary enhanced model arepresented in the last part.

2 NUMERICAL MODEL

2.1 Original DEM-PFV Coupled Model

Our DEM approach defines the mechanical properties of the interaction between grainswhose shape is assumed to be spherical. Following Newton’s laws, the positions of particlesare updated and calculated at each time-step of the DEM simulation. As introduced in[4], the PFV formulation is based on a simplified discretisation of the pore space as anetwork of regular triangulation and its dual Voronoi graph (1).

This network simplifies the formulation and resolution of the flow problem. The con-tinuity equation is expressed for each pore, linking the rate of volume change of one

tetrahedral element˙V fi to the fluxes qij through each facet. Each flux can be related to

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Figure 1: Regular triangulation (left) and Voronoi graph (right).

the pressure jump between to elements via a generalised Poiseuille’s law, so that

˙V fi =

j4∑j=j1

qij = Kij(pi − pj) (1)

couples the particles velocity to the fluid pressure field. The expression of conductivityKij has been validated recently by comparisons with glass beads experiments [10].

The total force exerted by the fluid on particle k can then be defined as [5]:

F k =

∫

∂Γk

ρkΦ(x)nds+

∫

∂Γk

pnds+

∫

∂Γk

τnds (2)

= F k,buoyancy + F k,pressure + F k,viscous (3)

2.2 Lubrication Forces

As classical poromechanics, the original DEM-PFV model takes into account theisotropic part of the stress and strain tensors (pressure and divergence of solid phasevelocity) in the coupling (eq. 1 in our case). The contribution of the fluid to the bulkshear stress is de facto neglected. It is worth noting that the shear part of the coupling issimilarly lacking in discrete models inspired by the coupling equations of poromechanics,such as the continuum-discrete methods [15, 16].

In order to deal with sheared suspensions, another viscous contribution has to beintroduced for modeling the shear stress. Various ways may be used for this purposesuch as viscous forces obtained in the framework of the lubrication theory developed byVan Den Brule [11]. The shear lubrication force defined by equations 4 seems to be inconcordance with the formulation obtained by the Finite Element Method (the Stokessolver of Comsol is used for this purpose). Results of this comparaison performed on a

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simple configuration of a sphere rotating at a given angular velocity in a regular assemblyof identical particles, are shown in figure (2).

Figure 2: Comparaison of viscous shear forces for the case of rotating sphere in a regular assem-bly of particles. h is the surface-to-surface distance and r is the particles radius.

F lubshear =

πµ

2(−2r + qln(

q

h))vt and F lub

normal =3

2πµ

r2

hvn (4)

µ is the fluid viscosity, r the particle’s radius, q the distance between particles,h is thesurface-to-surface distance and vt and vn are respectively the shear and normal velocityof the particle.

The normal interaction between two elastic-like particles in a viscous fluid is describedby the Maxwell visco-elastic scheme (figure 3) which combines a spring of stiffness kin series with a pad of viscosity η. This combination between the lubrication and theelasticity is close to that adopted by Rognon [9]. The force generated from this model isthen: F = kue = νuv where k is the model’s stiffness, ue is the elastic displacement, uv isthe viscous displacement (u = ue+uv is the total displacement) and ν is the instantaniousviscosity of the model defined, using the equation 4 as:

ν =32πr2µ

uv

.

By equalizing the elastic force and the viscous force, we have:

4

128


Figure 3: Visco-elastic scheme.

kue =

3

2πr2µ

uv

uv

=3

2πr2µ

(u− ue)

u− ue

=3

2πr2µ

(u− Fk)

u− F

k(5)

Using the finite difference method, F can be written as F = (F+ − F−)/∆t (F+ =F (t+ dt) and F− = F (t)). By substituting this relation into equation 5, we obtain:

F+ =3

2πr2µ

(u− F+ − F−

k∆t)

u− F−

k

(6)

When we express F+ as a function of F−, the normal lubrication force is:

F = F+ =ν(u+

F−

k∆t)

1 +ν

k∆t

(7)

2.3 Periodic Boundary Conditions

As the system is considered infinite in the flow direction, some problems can arise fromthe boundary effects in the numerical simulation. In order to avoid such problems, periodicboundary conditions are implemented in the PFV model (figure 4) (the periodicity forthe DEM part was developped independently [13]). Denoting by S = [s1, s2, s3] (figure 4)

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the period size in the three dimensions and by i ∈ N3 is the distance between one pointof coordinates r and its periodic image r’ = r+ S · i in an adjacent period, then the porepressure is expressed as follow:

p′ = p+�p · S ∗ i (8)

Figure 4: 2D periodic cell

3 NUMERICAL RESULTS

We generate an assembly of N=500 frictional grains of average diameter d = 0.14±0.03,density ρ = 2500 and friction angle Φ = 30. The assembly (Figure 5) is H=13d high,L=9d large and l=9d wide. The granular material is first confined under a constantpressure P, then sheared without gravity, between two parallel walls distant from H andmoving at a velocity ±V/2 respectively. In order to satisfy a quasi-static regime, V ischosen to be small so that the inertial number I = dγ

√ρ/P is less than 10−3.

.Simulations of such an assembly saturated of an incompressible fluid of viscosity µ

and submitted to a simple shear with shear rate γ = dVdH

at imposed vertical pressure P

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Figure 5: Simulation cell. The boundary conditions are: ux = V/2, σyy′ = P , uz = 0 and fluidpressure is zero. Periodic boundary conditions are applied in the x and y directions.

are presented. The viscous stress is decomposed in different contributions which can beexamined separately: contact forces, lubrication forces and drag forces associated to theporomechanical coupling.

Figure 6 shows the different contributions of each force applied on the granular media.Contrary to what is sometimes postulated in the litterature, our numerical results showthat tangential lubrication forces are significant compared with the normal ones. Thetotal shear stress applied on the material, being dry in the first stage, increases twicewhen the normal lubrication force is applied. The shear lubrication force, added after, isnot negligeable; it participates with around 80% of the normal contribution. This resultis interesting as tangential lubrication forces usually neglected compared with the normalones (e.g. Rognon et al [9])

4 CONCLUSIONS

In this contribution, we presented a new hydromechanical coupled model able to de-scribe the behavior of dense granular materials subjected to a shear flow under constantpressure. Contrary to what is sometimes postulated in the literature, the tangential lu-brication forces are significant compared with the normal ones. The proposed numericalmodel is able to describe the behavior of dense suspensions; the friction and dilatancylaws µ(Iv) et Φ(Iv), respectively, will be compared to experiments and rheological modelfrom the litterature [2] in future work.

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Figure 6: Shear stress applied on the granular media function of the shear rate. In a first stage wegenerate a dry material. Normal lubrication forces are added when 40% of skeleton’sdeformation are reached. Shear lubrication forces are applied then. Finally, pressureforces associated to the poromechanical coupling are applied.

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[16] J. Zhao and Shan T. Coupled CFD–DEM simulation of fluid–particle interaction ingeomechanics. Powder Technology, 239(0):248 – 258, 2013.

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